The sulfur generated by the garlic plant serves as an all natural fungicide, helping to suppress many fungal related plant diseases.
Mint serves as an insect repellent and is a good companion plant for beets.It deters many moths such as cabbage moth, Beetles and fleas, aphids. Smaller rodents such as moles and field mice. Mint is known to attract some beneficial insects such as predatory wasps.
Mint is believed to improve the quality of cabbage [unproven] Cabbage and Brassicas are also good companions for beets making it feasible to grow all three together.
One drawback to mints is that they are incredibly invasive and if not well maintained will dominate the area they are planted in.
Bush Beans, such as butter beans, limas, soy, green beans grow well with Beets. Bush beans also add nitrogen to the soil, an added plus , coupled with the magnesium from the beets, the soil should be in good shape for the next crop to grow there.
For reasons unknown Beets do not fare well with pole beans or runner beans as they inhibit each others growth.
"Beets grow well near bush beans, onions or kohlrabi but are turned off by pole beans. Field mustard and charlock also inhibit the growth of beets. Lettuce and most members of the cabbage family are "friendly" to them. " Carrots Love Tomatoes - L. Riotte
Onions are compatible with beets. They draw from different levels in the soil, so do not have to compete for nutrients. Some varieties of Onions and related alliums such as garlic help to suppress many fungal related plant diseases.
All Brassica plants which includes cabbage, broccoli, brussel sprouts, kale, cauliflower, kohlrabi are compatible with beets. Lettuce, although not actually a Brassica also goes well with beets. The leafy veggies benefit from the added magnesium provided by beets and there are no drawbacks for the beets.